Kinetic energy indicates the energy possessed by the object because of its motion or velocity, thus the term kinetic energy directly depends on the velocity of the object. If the velocity and mass of the object are known, then it becomes easier to calculate the kinetic energy of the object by using the relation `\frac{1}{2}`.mV^{2}.

**But if the velocity of the object is unknown then also we can find the kinetic energy of the object based on what else data is available to us.**

Following are the different cases in which we can find the kinetic energy of the object without knowing the velocity of the object:-

Contents:

## From momentum of the object:

** If the momentum ‘P’ and mass ‘m’ of the object are given**, then the kinetic energy of the object can be calculated by using the following formula.

KE = `\frac{P^{2}}{2m}`

## By work-kinetic energy theorem:

** If the force acting on the object and the displacement of the object is given** then the kinetic energy can be calculated by using the work-kinetic energy theorem. In this case, we can find the kinetic energy of the object without knowing the velocity and mass of the object.

As per the work-kinetic energy theorem, the change in kinetic energy of the object is equal to the net work done by the forces onto the object.

W = `\Delta` KE

∴ W = KE_{Final} – KE_{Initial}

Where the work done on the object is given by,

W = Force *×* displacement = F.x

Thus by knowing the forces acting on the object and the distance, it becomes easier to find the changes in the kinetic energy of the object.

**Example:-**

As shown in the below figure, the man is pushing a block on a frictionless surface with a constant force of 20 N over a distance of 3m in a linear direction.

The work done by the man on the object is given by,

W = Force (F) *×* Displacement (x) = 20 *×* 3 = 60 J

Thus as per the work-energy theorem, if the object starts from rest, the final kinetic energy of the object is given by,

W = KE_{Final} – KE_{initial}

60 = KE_{Final} – 0

KE_{Final} = 60 J

## Conservation of mechanical energy:

As per the principle of conservation of mechanical energy, the sum of the potential energy and the kinetic energy remains constant.

KE + PE = Constant (E_{Mechanical})

Where the constant indicates the total mechanical energy (E_{Mechanical}) associated with the object.

**Following are some of the different cases in which we can use the above principle to find the kinetic energy of the object:-**

### From gravitational potential energy:-

In this case, ** if we know the mass and the height of the object**, then we can calculate the kinetic energy of the object.

The below figure shows the object free-falling from the height of H_{max}.

The kinetic energy of the free-falling object, at a height of y (KE_{y}) from the ground, is given by,

KE_{y} = E_{Mechanical} – PE_{y}

Where, PE_{y} = Potential energy at height y

In this case, the E_{Mechanical} is equal to the maximum potential energy or the initial potential energy of the object. Initially, due to the height of H_{max}, the object possesses maximum potential energy (mgH_{max}) and zero kinetic energy,

Thus the above equation becomes,

KE_{y} = PE_{Max} – PE

As PE_{Max} = mgH_{Max} and PE_{y} = mgy,

∴ KE_{y} = mgH_{Max} – mgy

### From elastic potential energy:-

The below figure shows the mechanism used in the toy gun. Here the compressed spring is used to propel the ball.

In this case, the potential energy of the compressed spring is converted into the kinetic energy of the ball.

The potential energy stored in the spring of spring constant (K) due to the x amount of compression is given by,

PE_{spring} = `\frac{1}{2}`.Kx^{2}

Thus during the propel of the ball, the potential energy of the spring is converted into the kinetic energy of the ball.

Thus as per the principle of conservation of mechanical energy, after releasing the spring, the kinetic energy of the ball is given by,

KE = PE_{spring} = `\frac{1}{2}`.Kx^{2}